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```									            The Pigeonhole Principle

9/18/2012                              1
The Pigeonhole Principle

• In words:
– If n pigeons are in
fewer than n
pigeonholes, some
n
pigeonhole must
contain at least
two pigeons

What is n?
http://www.blog.republicofmath.com/archives/3115

2
9/18/2012
The Pigeonhole Principle

• In math:
Let f : A  B, where A and B
are finite sets and A  B .
Then there exist distinct elements
a1 ,a2 A such that f (a1 )  f (a2 ).

3
9/18/2012
The Pigeonhole Principle
Let f : A  B, where A and B
•   What is a set?
are finite sets and A  B .
•   a finite set?       Then there exist distinct elements
•   What is |A|?        a1 ,a2 A such that f (a1 )  f (a2 ).

•   What is a function?
•   the domain of a function?
•   the codomain of a function?
•   Why say “distinct”?

4
9/18/2012
Applications of The Pigeonhole Principle

• In any group of 8 people, two were born
on the same day of the week
• What are the “pigeons” and what are the
“pigeonholes”?
• A = the set of people, B = {Sun, … Sat},
f(a) = the day of the week on which a was
born

5
9/18/2012
Applications of The Pigeonhole Principle

• Suppose each pigeonhole        D   D   D   D   D
contains one bird, and         D   D   D   D   D
every bird moves to an
D   D   D   D   D
left or right). Show that no   D   D   D   D   D
matter how this is done,       D   D   D   D   D
some pigeonhole winds up
with at least 2 birds.

6
9/18/2012
Applications of The Pigeonhole Principle

• Suppose each pigeonhole        D   D   D   D   D
contains one bird, and         D   D   D   D   D
every bird moves to an
D   D   D   D   D
left or right). Show that no   D   D   D   D   D
matter how this is done,       D   D   D   D   D
some pigeonhole winds up
with at least 2 birds.

7
9/18/2012
Applications of The Pigeonhole Principle

• Suppose each pigeonhole          D     D    D     D       D
contains one bird, and           D     D    D     D       D
every bird moves to an
D     D    D     D       D